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Preprint submitted on 10 Feb 2010

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Modelling collision outcome in moderately dense spraysGautier Luret, Thibaut Menard, Gregory Blokkeel, Alain Berlemont, Julien

Reveillon, François-Xavier Demoulin

To cite this version:Gautier Luret, Thibaut Menard, Gregory Blokkeel, Alain Berlemont, Julien Reveillon, et al.. Mod-elling collision outcome in moderately dense sprays. 2010. hal-00455627

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Submitted to International Journal of Atomization and Spray as a full length paper, selected among ILASS–Europe 2008 conference papers Modelling collision outcome in moderately dense sprays Gautier LURET, Thibaut MENARD, Grégory BLOKKEEL, Alain BERLEMONT, Julien REVEILLON and François-Xavier DEMOULIN

Gautier LURET, Thibaut MENARD, Alain BERLEMONT, Jul ien REVEILLON, François-Xavier DEMOULIN Coria UMR6614 CNRS Université de Rouen BP 12, Site universitaire du Madrillet 76801 Saint Etienne du Rouvray Cedex, France Grégory BLOKKEEL PSA Peugeot Citroën. Route de Gisy 78943 Vélizy-Villacoublay Cedex, France Author correspondence : François-Xavier DEMOULIN Coria UMR6614 CNRS

Université de Rouen BP 12, Site universitaire du Madrillet 76801 Saint Etienne du Rouvray Cedex, France

Tel : 00 33 2 32 95 36 74 Fax : 00 33 2 32 91 04 85 E-mail : demoulin@coria.fr

2

Abstract

To simulate primary atomization, the dense zone of sprays has to be addressed and new

atomization models have been developed as the ELSA model [4]. A transport equation for the

liquid/gas interface density is stated and extends the concept of droplet diameter. Several

related source terms require modelling attention. This work describes the contribution of

collision and coalescence processes. Several questions arise: Is it possible to represent

collision/coalescence from an Eulerian description of the flow? What are the key parameters?

What are the particular features of collision in dense spray? To answer these questions, a

Lagrangian test case, carefully resolved statistically, is used as a basis to evaluate Eulerian

models. It is shown that a significant parameter is the equilibrium Weber number: If it is

known, Eulerian models are able to reproduce the main features of Lagrangian simulations.

To overcome the Lagrangian collision model simplification that mostly considers collisions

between spherical droplets, a new test case has been designed to focus on collision process in

dense spray. The numerical code, Archer, which is developed to handle interface behaviours

in two-phase flow by the way of direct numerical simulation (DNS) [19] is used. Thanks to

DNS simulations and experimental observations, the importance of non spherical collisions is

demonstrated. Despite some classical drawbacks of DNS, we observed that an equilibrium

Weber number can be determined in the considered test case. This work emphasizes the

ability of interface DNS simulations to describe complex turbulent two phase flows with

interfaces and to stand as a complement to new experiments.

3

1 Introduction

Energy and environment survey are becoming public policy issues, such as greenhouse effects

and global warming. As a consequence, drastic limitations of emissions are imposed to the

automotive industry. The European car manufacturers will be subjected to more stringent

emissions regulation (such as Euro 5 and post-Euro 5 standards), concerning nitrogen oxides

(Nox), CO2, unburned hydrocarbons (HC) and particle emissions. To reach these tolerance

thresholds, the automotive research is looking for multiple processes that are involved in

combustion: fuel distribution (injection), internal aerodynamics (vaporization, mixing) and

combustion itself (ignition, chemical reactions).

For many decades, great attention has been devoted to the injection process. It is a

determining factor for the fuel distribution inside the combustion chamber, and it contributes

indirectly to the pollutants formation. Its optimization may notably lead to a cleaner Internal

Combustion Engine (ICE). For instance, following the injector nozzle geometry [1] and the

injection strategy [2, 3], the overall spray behaviour and its characteristics may be drastically

different. The liquid jet atomization has thus to be well-understood, but unfortunately several

complex mechanisms are involved, such as turbulence, primary and secondary breakup,

droplet collision and coalescence. All these mechanisms have to be taken into account to

determine the spray dispersion and the local droplet diameter and velocity distributions. From

a correct modelling of the injection and atomization process it is then possible to compute in a

realistic way the droplet-vaporisation, vapour mixing and combustion processes [4].

Furthermore, one characteristic of high-pressure Diesel jets is the presence of a liquid core

which is attached to the exit of the nozzle. This part of the jet and its vicinity are more

commonly called the dense zone. It is the most difficult zone to model partly because of the

lack of experimental data despite recent advances in optical techniques [5-7]. However it is

4

obvious that a realistic description of the dense part improves significantly the global

modelling of the injection [4, 8-10]. To deal with the dense part, Lebas et al. [4] extended the

Eulerian Lagrangian Spray Atomization (ELSA) model (originally proposed by [11]) to

Diesel injection. The model is based on a single-phase Eulerian description of the flow that is

composed of a liquid and a gas mixture. The initial dispersion and atomization of the liquid jet

are assumed to be dominated by the turbulence, but by taking into account for high variable

density [12]. A transport equation for the mean liquid/gas interface density is also considered

to describe the complex liquid topology. Indeed, in the initial part of the jet, the notion of

droplet diameter is not applicable, as no droplet is formed yet. Thus the quantity of interface

is a first order parameter that can help in describing the different interactions between the

liquid and the gas phases.

The surface density is a particular variable because it can be defined locally only by using

generalized functions. This is a Dirac function. Nevertheless, the local equation can be

determined, see for instance a review on this problem by Morel [13]. But the link has not yet

been established with modeled equations that are currently used for RANS (Reynolds

Average Navier Stokes) simulations [4, 11, 14] or LES (Large Eddy Simulation) simulations

[15] . Among the processes that play a role on the surface density, the effect of

collision/coalescence is expected to be significant especially because we deal with the dense

zone of the spray. Different terms in the currently used equations are assumed to take the

collision/coalescence effects into account. But it remains obvious that extended researches on

this topic are still required.

The first question concerns the dispersed phase in the resulting spray and the ability of

Eulerian methods to represent collision/coalescence phenomena. Indeed by comparisons with

Lagrangian methods that totally describe the PDF (probability density function) of the spray,

the Eulerian approaches are generally using only few moments of the distribution to describe

5

the whole PDF. As a consequence, information can be lost, depending on the assumed shape

of the PDF, that can only be built with the retained moments. The collision process describes

how two or more droplets interact when their respective motion induces crossing trajectories.

It is obvious that characteristics of each colliding droplet are important to determine the

regime of the collision, see for instance [16]. Clearly, knowing the outcome of each individual

collision is not among the capability of Eulerian methods but this is not necessary. The

expected behaviour of the collision/coalescence model for an Eulerian method is to predict

correctly the effect of the whole set of collision on the retained statistical moments used to

describe the spray. Considering the Eulerian methods currently used to describe the

atomization, this property will be studied as far as the mean surface density is concerned. To

do so, a test case, well resolved from a statistical point of view using Lagrangian models of

collision/coalescence, is proposed.

The second problem concerns the validity of the collision/coalescence models currently used

to compute the evolution of the mean (or filtered) surface density in the dense zone. Indeed,

the first class of models proposed by Vallet et al. [11, 17] and then by Iyer and Abraham [18]

are based on droplet collisions. Therefore, these models are applied in the dense part of the

spray where the droplets are not formed yet! A global model proposed specifically for the

dense part of the spray has been proposed by Lebas et al. [4]. Although, this model

participates to the correct behaviour of the computed spray, it has not been proved to behave

properly as far as the collision/coalescence processes are concerned. Moreover it has been

outlined [4] that some of its parameters have still to be established. Collision has been studied

experimentally only for droplet collision, conditions that are far to those encountered in the

dense part of the spray. To design and to characterise an experiment of collision between two

non spherical parcels of liquid is not an obvious task. Consequently to understand collisions

in the dense zone, a numerical test case has been put forward thanks to the Archer code that

6

has been developed to compute two-phase flow by the way of direct numerical simulation

(DNS) [19]. These simulations are still difficult and computationally expensive, thus,

preliminary results are presented here. They concern few cases representative of what can be

encountered during an atomization process.

This article is organized as follow: In the first part the Eulerian approach ELSA used to

compute the atomization is described. Then a Lagrangian simulation is put forward to

determine the ability of the Eulerian approach to represent the collision/coalescence

phenomena. The third part of this paper describes a direct numerical simulation used to study

collision and coalescence of non spherical liquid parcel.

2 THE EULERIAN-LAGRANGIAN SPRAY ATOMIZATION MODEL In this section the ELSA model is described to understand where the collision/coalescence

effects are expected to play a role in the complete formulation. The goal of the ELSA model

is to describe realistically the dense zone of the spray. Based on the assumption that the

Weber and Reynolds numbers have to be high, the ELSA model is naturally well adapted to

Diesel Direct Injection conditions. This assumption corresponds to an initial atomization

dominated by aerodynamic forces. The global behaviour of the model and its ability to

describe Diesel injection have been checked out by Lebas et al. [4].

A liquid-gas flow is considered as a unique flow with a highly variable densityρ which can be

determined thanks to the following equation:

g

l

l

l YY

ρρρ

~1

~1 −

+= (1)

˜ Y l corresponds to the mean liquid mass fraction. While, gρ and lρ are respectively the gas and

the liquid densities. gρ follows the state equation of a perfect gas (by taking the liquid

7

volume fraction into account) and lρ account for the fact that the liquid density can be

modified accordingly to the liquid temperature.

Considering the two-phase flow as a unique mixture flow with a highly variable density

implies that the transport equation for the mean velocity does not contain any momentum

exchange terms between the liquid and the gas phases. Additionally, under the assumption of

high values of both Reynolds and Weber numbers, laminar viscosity and surface tension

forces can be neglected:

i

ji

ij

jii

x

uu

x

P

x

uu

t

u

∂′′′′∂

−∂∂−=

∂∂

+∂

∂~~~~ ρρρ

(2)

This “mixture” approach has to be combined with a turbulence model. The (k-ε) model is

generally used even if other models have been tested [12]. The Boussinesq hypothesis is

chosen to model the Reynolds stress tensor.

A regular transport equation stands for the mean liquid mass fraction Y l with a source term

representing the effect of vaporization:

Ω−

∂∂

=∂

∂+

∂∂ ~

~~~~

,,

ELSAvj

l

lt

t

ji

ill mx

Y

Scxx

uY

t

Y&ρµ

∂∂ρρ

(3)

Ω~ is the liquid-gas interface density per unit of mass and ELSAvm ,& represents the vaporization

rate per unit of mass. It has been modeled from Abramzon and Sirignano’s approach [20, 21].

To determine the amount of surface between the two phases, classical approaches consist in

considering spherical liquid drops and than using the diameter as geometrical parameter. But

a more general parameter has to be used where a diameter of droplet cannot be defined: the

8

liquid-gas interface density, notedΣ when expressed per unit of volume or Ω~ when given per

unit of mass. The following equation relates both definitions:

Σ=Ω~ρ (5)

The transport equation for this variable is postulated. In the latest version of the ELSA model,

it takes the following form [4]:

( ) ( )( ).2./...,

1~~~~

vapondBUcoalcollturbinitjt

t

jj

j

xScxx

u

tφφφφφµ

∂∂ρρ ++Ψ−++Ψ+

∂Ω∂=

∂Ω∂

+∂

Ω∂

Ω

(6)

This equation must be applicable from the dense zone up to the dispersed spray where

droplets are eventually formed. In this latter case, an equivalent diameter of Sauter can be

defined using the liquid-gas interface density and the mean liquid mass fraction :

D32 = 6 ˜ Y lρ l

˜ Ω (7)

Each source termφi (equation 4) models a specific physical phenomenon encountered by the

liquid blobs or droplets. Lets

φ init . = 12ρ µt

ρ l ρgSctLt

∂ ˜ Y l∂xi

∂ ˜ Y l∂xi

(8)

be an initialization term, taking high values near the injector nozzle, where the mass fraction

gradients take its highest values. It corresponds to the minimum production of liquid-gas

interface density necessarily induced by the mixing between the liquid and gas phases, see

Beau and Demoulin [22].

ΩΩ−Ω=

*1

. ~

~1

~

tturb τ

ρφ (9)

9

φturb.corresponds to the production/destruction of liquid gas interface density due to the

turbulent flow stretching and the effects of collision and coalescence in the dense part of the

spray. It is supposed to be driven by a turbulent time scaleτ t . This production/destruction

term is defined to reach an equilibrium liquid-gas interface density *1

~Ω . It corresponds to the

quantity of surface obtained at equilibrium under given flow conditions. Several formulations

can be proposed. In Lebas et al. [4], an equilibrium Weber number is supposed: 1*1 =We :

*1

*1

~~~

We

Yk

ll

l

σρρ

=Ω (10)

Where σ l is the surface tension of the liquid phase.

φcoll./ coal.models the production/destruction of liquid-gas interface density due to the effects of

collision and coalescence in the dilute spray region. Different proposals will be discussed

extensively in the next section of this article.

ΩΩ−Ω= 0;~

~1

~

*32

2ndBU

ndBU Maxτ

ρφ (11)

φ2ndBU deals with the production of liquid-gas interface density due to the effects of secondary

breakup in the dilute spray region. This source term is derived from the work of Pilch and

Erdman [23]. It enables the estimation of the breakup time scaleτ 2ndBUaccordingly to the

Weber number of the gas phaseWeg, thanks to empirical correlations. Moreover, it determines

the stable Weber number *3We with:

( )6.1*3 077.1112 OhWe += (12)

Oh is the Ohnesorge number. The stable interface density that corresponds to stable droplets

as far as the secondary breakup is concerned, written as follow:

˜ Ω crit ,3 = 6ρ l urel2 ˜ Y l

ρ lσ lWecrit ,3

(13)

10

with urel the relative velocity between the liquid and gas phases. Vaporisation is characterized

thanks to:

ELSAv

l

vapo mY

,

2

~

~

3

2&

Ω−= ρφ (14)

φvapo.comes from a classical adaptation of the “D2” law of vaporization models for droplets

and deals with the effects of destruction of liquid-gas interface density due to vaporization.

The transport equation of ˜ Ω takes into account several physical phenomena encountered by

the liquid phase. Some of them are specifically observed in the dense zone of the spray and

other are dedicated to dispersed spray regions. A functionΨ has been introduced to switch

from the dense formulation to the dispersed formulation continuously and linearly in term of

liquid volume fraction [4]. The transition zone is determined by two volume fraction limit

values: 5.0=denseφ and 1.0=diluteφ . The liquid volume fractionφl can be obtained thanks to the

following relation:

φ l = ρ ˜ Y lρl

(15)

This description of the ELSA model show how an Eulerian method can be derived to deal

with atomisation. Though it has been shown [4] that the presented form of the model is able to

capture the global features of the atomisation of a Diesel jet, more detailed studies are still

required. Indeed, to validate the various source terms of the surface density equation, specific

studies for each kind of phenomena are required. In the following, test cases are put forward

to study the particular effect of collisions. This phenomenon is expected to be important for a

model devoted to the dense zone of a spray.

3 Variation of surface by collision in Eulerian approaches Two types of processes due to collision can substantially modify the liquid-gas interface

density with particularly opposite effects on its development.

11

On the one hand, coalescence decreases the surface density. Indeed, when two droplets merge

into one, the surface area between the liquid and gas phases decreases. On the other hand, the

collision-induced break-up plays the role of a production term and it has obviously a reverse

effect. Several satellite droplets, produced after collision, represent a more important surface

area than their parent droplets. Consequently, if flow conditions are kept statistically

stationary, an equilibrium state is reached. It can be characterized thanks to a mean

equilibrium surface density *~Ω , which is related to an equilibrium Sauter mean diameter *32D

(Eq. (7)) and an equilibrium Weber number of collision, *We :

( )l

llcoll

DuDWeWe

σρ *

322

*32

* == , (16)

where lu is the liquid velocity fluctuation. Note that the equilibrium Weber number of

collision represents the ratio between liquid kinetic energy of agitation and the surface energy

of the spray. It is different of the collision Weber number used to characterize each collision

between two droplets. This equilibrium state characterizes the asymptotic behaviour of a spray

that experiences collision processes. Nevertheless, to properly describe these processes, it is

necessary to predict the behaviour of the spray up to its equilibrium state. From an Eulerian

point of view, several formulations of the collision source terms in liquid surface density

equation have been proposed .

Iyer and Abraham [14] derived an expression adapted from a Lagrangian approach, initially

developed by O’Rourke and Bracco [24] and based on the collision frequency between liquid

droplets:

2

22 d

ldcoll

dcoll

DuN

Nf

πτ

== (17)

Where, lu is an approximation of the relative velocity between the droplets. It depends on the

liquid kinetic energy:

12

3

2 lcoll

kCu = (18)

The decrease rate of the number density due to the coalescence depends on a coalescence

probability or coalescence efficiency collη derived from Brazier-Smith et al.’s correlation

[25], which can be written as:

collcolld f

dt

dN η−= (19)

With,

= 1;

24.6min

collcoll We

η (20)

The collision source term derived for the surface density equation, Eq.(6), has been adapted

from the Lagrangian formulation and retained by Iyer and Abraham [14]. It is given by:

12~ 2

./.l

collcoalcoalcoll

uΩ−== ρηφφ (21)

Unfortunately, Iyer and Abraham did not propose any source term concerning the collision-

induced breakup, even if other source terms counterbalance the coalescence in their case.

Source terms are due to aerodynamic breakup and to vaporization. But if collisions are

considered on their own, the diameter will diverge inexorably towards an infinite value (see

Fig. 1). This model is referred as “Iyer” in the following.

13

Within the ELSA framework presented previously Beau and Demoulin [22] have suggested

one source term for each process (coalescence and collision-induced breakup) as did Vallet et

al. [11]:

ΩΩ−Ω=⇒

Ω=

ΩΩ−=

2*/

2*

2

~1

~

~

~

collcoalcoll

collbup

collcoal

τρφ

τρφ

τρφ

(22)

The form of the characteristic collision time scale collτ is similar to Iyer and Abraham but

written with variables available within the ELSA framework:

k

C

uS

L col

leff

collcoll

~

3

2~

3

Ω==

ρτ (23)

If one considers a collision between two droplets characterized by a Weber number collWe .

Then, for an initial surface density Ω~ , the equilibrium surface density 2*Ω given by

conservation of the total energy becomes:

61

61

~

*

2*

collWe

We

+

+Ω=Ω , (24)

where *We is the equilibrium Weber number. Its value has been initially set to 15* =We [22].

Indeed, this is the approximate value that separates coalescence and separation effect after

collision [16]. This model is referred as “Beau” in the following.

From an experimental point of view, binary collision outcomes have been extensively studied,

see for instance [16, 26]. Two main parameters are generally used to build a diagram of

collision depending on the Weber number of collision together with the impact factor B .

Nevertheless, the equilibrium value of the Weber number is not a direct output of these

14

diagrams. Collision results in new droplets which will experience an extra collision, the

missing information is what will be the next Weber of collision after the considered collision.

To determine the equilibrium Weber number, another possibility is to find out the distribution

of the total energy between the surface energy and the kinetic energy. For instance, at the

equilibrium, if the liquid kinetic energy balances the liquid surface energy, then the following

relation is obtained:

12~

~6

2

1 2*2 =

Ω=⇒=

l

llllll

YuWeSum

σσ (25)

This model of collision/coalescence effect with this value of *We has been used in the last

version of the ELSA model [4].

However, the formulation of the collision/coalescence source term of Eq.(22) has been

postulated but not demonstrated. In this paper, in order to overcome this weakness, a new

source term, inspired from Iyer’s approach, but with an addition of the contribution due to

collision-induced breakup is proposed:

( )3

~2 2 l

bupcoal

uΩ−=+ ρβφ (26)

whereβ corresponds to the number of droplets formed by the collision between two parent

droplets. Ifβ is equal to 1, then this source term is a destruction term of the liquid-gas

interface due to coalescence and it is equivalent to the one of Iyer and Abraham. If β is

greater than 2, the source term becomes a production term of interface due to liquid breakup.

From an Eulerian point of view, droplet collisions are considered as a set of collisions and not

individually. Accordingly, β does not take necessarily an integer value. Due to collisions the

spray evolves in order to reach the equilibrium diameter *32D . If the current mean diameter

15

32D is lower than *32D , then coalescence is expected in mean, otherwise collision induced

breakup is expected:

>

∈

≤

∈

*32323*

32

332

*32323*

32

332

2;2

2;1;2

max

DDifD

D

DDifD

D

β

β

(27)

Since the probability density function of the variableβ for individual collision is not known,

a uniform distribution is used as a first step. Thus :

2maxmin βββ +

= , (28)

where, [ ]maxmin ,ββ describes the prescribed interval of β , see Eq. (26). This model is

referred as “New” in the following.

To study the behaviour of each model, a simple test case is put forward. It consists in a

spatially homogeneous spray where the kinetic energy of liquid agitation is kept constant. In

this case, all parameters of the models are constant. The only variation is due to the evolution

of surface density. This can completely be described by a unique Weber number of collision.

The previous three models are tested on Fig. 1. Beau’s model and New’s model take into

account both the coalescence and the collision induced breakup. Accordingly, after a while

they tend to a constant Weber number. At the contrary with the Iyer’s model, the Weber

number still increases all along the simulation. Equilibrium is never achieved since only

coalescence is considered as far as collisions are concerned. For the three models, the

characteristic time is identical but each model leads to a different behaviour during the

coalescence phase. It appears that the law proposed by Vallet et al. [11] is not in agreement

with the other models that are based on droplet interactions. For pure coalescence, the Iyer’s

16

model and the New’s model are identical except for the initial value of 12, proposed by Iyer

and Abraham [18] that becomes 3 in the New’s model.

(Figure 1: about here)

These Eulerian models are compared to their Lagrangian counterpart in the next section.

4 Collision/coalescence: Lagrangian point of view

To model a spray, the traditional Lagrangian approach based on the Discret Droplet Model

(DDM) [27], consists in a statistical description of the spray. Stochastic particles or parcels,

group of real droplets with similar properties (diameter, temperature, velocities), are tracked in

a Lagrangian way inside the domain. To represent accurately a spray, a large particle sample is

required. This is one of the major drawbacks of this approach that need additional

computational efforts. Unfortunately, collision submodels do not escape to this rule. They are

even more expensive in terms of computational resources than the other modeled phenomena.

Thus, a compromise between the statistical convergence and the computational resources has

to be found.

Two questions are inherent to Lagrangian collision submodels:

1. How to determine the occurrence of collision?

2. How to determine collision outcome?

The first question is merely a numerical and mathematical question. It consists in determining

the collision partners and the probability of collision.

Several algorithms exist. A widely known model is without doubt the O’Rourke’s model [24]

that is based on the kinetic theory through the calculation of a collision frequency.

17

Unfortunately, this algorithm has several limitations. In particular it can lead to mesh-

dependent results and it may have a prohibitive computational cost. Several authors have done

proposals to overcome to these problems.

Nordin [28] transposed the collision process in terms of intersection of parcel trajectories. So,

droplet collisions have been restricted to parcels whose trajectories intersect at the same time

during the time step. To avoid considering all the possible collision partners, even the most

unlikely, two criteria related to the parcel displacement have to be respected. Schmidt and

Rutland [29] extended the No-Time Counter (NTC) algorithm based on a pre-sampling of

collision partners. They used a second independent mesh that is specifically dedicated to the

collision process. Li et al. [30] proposed an algorithm based on the Smoothed Particle

Hydrodynamics (SPH) method. For each parcel, only the closest parcels are considered to be

likely to collide. A collision probability defined from the kernel function of the relative

distance is then used to determine whether collision occurs.

The second question concerns the collision outcome.

When a collision between two droplets occurs, it is necessary to forecast the collision outcome

and to know the post-collision characteristics. The collision outcome depends on the properties

of both phases and the intrinsic parameters of the collision (relative velocity, impact parameter

and size ratio). In this work, only binary collisions are considered. Several regimes have been

observed, such as bounce, permanent coalescence, coalescence followed by a separation

(reflexive or stretching) and accompanied or not by new satellite droplets [16]. The boundaries

between these regimes are now relatively well-known for low ambient pressures. Recently, a

particular attention was devoted to the satellite droplet formation [31, 32] see for instance Fig.

2. This attention is quite legitimate, because satellite droplets formation is very likely to occur

in the spray dense region.

18

(Figure 2: about here)

However, the development of a collision model able to predict the number of satellite droplets

along with their sizes and velocities, knowing the medium and liquid properties and the

collision characteristics is not fully achieved.

Georjon and Reitz [33] suggested a simple model to translate observations made about the

satellite droplets creation for high Weber number. When two droplets collide, a cylinder-shaped

liquid mass is induced by the impact, see Fig.2. Then instabilities may propagate, break this

liquid ligament and form several satellite droplets. This shattering-collision process takes place

beyond a specific value of collision Weber number chosen somewhat arbitrarily, namely about

one hundred. Post and Abraham [34] studied collision phenomena for Diesel sprays and

proposed a complete model accounting for the above-cited regimes [16] based on experimental

results. Because of the computational cost, they simplified the model of Georjon and Reitz for

shattering collisions for high Weber values and, they took the effects of local pressure into

account: when pressure increases, the domain of bounce regime is enlarged. Hou and Schmidt

[35] used the simplification done by Post and Abraham, but considered only the interaction

volume of colliding droplets to produce the satellite droplets.

To focus on collision processes, Orme [36, 37] studied the binary droplet collisions in vacuum

so that no other process (for instance gas turbulence or secondary break-up) interferes.

Similarly, as a test case, collision processes are simulated without taking the action of the gas

phase on liquid droplets into account. The droplets are considered ballistic. This avoids the

complex interactions between gas and liquid turbulence that depends on turbulent scales and

inertial time. The computational domain is cubic box with periodic boundary conditions in all

direction. A Lagrangian method is used to follow the stochastic particles, Fig. 3.

19

(Figure 3: about here)

In this configuration, statistical convergence can be reached by using a sufficiently large

sample of stochastic particles. The test case presented here is identical to the one used

previously to compare Eulerian models. The configuration is simple enough to allow the

O’Rourke’s approach [24] to be used, since only one mesh cell is necessary as far as the

Lagrangian simulation is concerned. To compare with the Eulerian cases, the mean liquid

kinetic energy is forced to be constant. Otherwise, due to the coalescence phenomenon, a large

part of the initial liquid kinetic energy will be dissipated, see Fig. 4. When coalescence

phenomenon occurs, parents droplets have not initially the same velocity. But the resulting

drop has only one velocity, hence a part of the initial kinetic energy carried by the parent

droplets has vanished. In reality, the dissipated energy creates oscillations within the child

droplet. To compensate this sink of liquid kinetic energy, two kinds of forcing have been

tested:

1. A linear forcing [38] originally used for the gas phase is applied to the liquid by adding a

source term to the droplet velocity equation:

ii Au

dt

du+= .... (29)

Where, A is a parameter continuously determined during the computation to compensate

exactly the dissipated energy.

2. A complete redistribution of the droplet velocity field at each time step to maintain a

prescribed level of liquid kinetic energy.

20

(Figure 4: about here)

In this study, both forcing techniques give similar results (see Fig. 4). Results presented in the

following are obtained using the second method. A consequence of this forcing technique is

that it erases any influence of the coalescence on the turbulent liquid field. Accordingly, this

effect is not studied here and the present study focuses on collision effects on droplet surface

variation.

Initially, the droplets are randomly distributed in the cubic box with the same diameter. Five

thousand stochastic particles have been retained to achieve statistical convergence.

Characteristics of the test case are summarized in table 1:

(Table 1: about here)

Concerning the collision outcomes considered in this study, the coalescence and stretching

separation regimes will be taken into account through the Brazier-Smith’s correlation [24, 25].

This model is referred as “O’Rourke” in the following. To account for the breakup induced by

shattering collision the model of Georjon and Reitz [33] is used and referred as “Georjon”.

Finally, the most complete model tested in this study is the one proposed by Post and Abraham

[34] and is referred as “Post”.

Results are presented as a function of the non-dimensional time obtained by using the collision

time scale as a reference. A comparison of the three Lagrangian models tested in this work is

presented in Fig. 5. O’Rourke model does not account for any collision-induced breakups.

Accordingly the Weber number increases all along the simulation. This evolution corresponds

21

to a continuous increase of the droplet diameter. However, the efficiency of coalescence

decreases with time as higher values of the Weber number are reached. Then, the increase rate

of the Weber number becomes smaller and smaller, but no equilibrium value is found.

Paradoxically, both Georjon and Post models include collisions induced breakup effects and

lead to an equilibrium Weber number. The Georjon model behaves similarly to the O’Rourke

model at the beginning since droplets are very small and no breakup is expected. Then the

simple formulation used in Georjon model to represent breakup due to shattering collision

switches on and nearly immediately an equilibrium Weber number ( 12* =We ) is found. The

complete formulation of the Post model takes into account more physical phenomena and

balance between coalescence, rebound and breakup is more complex. Thus the differences with

the O’Rourke model appear earlier and the transition to the equilibrium Weber number

( 15* =We ) is smoother. Notice that for both Georjon and Post models, equilibrium Weber

number values found in these simulations are not part of the model parameters. A computation

of the balance between the various phenomena taken into account by each model is required to

determine each corresponding equilibrium Weber number. Values found by Georjon and Post

models are different. This is expected since these models are not equivalent. Moreover, these

models have been designed with the goal to be in accordance with phenomena observed in

binary collisions rather than to fit an experimental value of the equilibrium Weber number.

Indeed, such a value has not been measured at our best knowledge and it will not be an easy

task. It has to be said that the equilibrium Weber number value depends on the model

parameters. Results presented here used standard parameters found in the literature [33, 34].

But some of them are not totally established, for instance Luret et al. [39] have shown variation

of the equilibrium Weber number when varying the shattering collision Weber number of the

Georjon model. Finally, it is interesting to note that both Lagrangian models lead to quite

similar equilibrium Weber numbers. That are close to the values proposed in the Eulerian

22

formulation discussed previously.

(Figure 5: about here)

Fig. 6 presents comparisons of the Eulerian models with the Lagrangian Georjon model. To

realize these simulations, the equilibrium Weber number found for the Georjon model has been

used ( 12* =We ) for both Beau and New Eulerian models. Phenomena taken into account by

the Iyer model do not allow the equilibrium to be achieved. Both Eulerian models are able to

represent correctly the evolution of the spray as far as the simple Georjon model is concerned.

Initially, the New model is closer to the Georjon model than to the Beau model. This is

expected because the Eulerian New model has been built considering binary collisions as it is

the case in the Lagrangian formulation.

(Figure 6:about here)

Figure 7 shows a comparison of the Eulerian models ( 15* =We ) with the more complete

Lagrangian Post model. The Eulerian models are not able to reproduce completely the complex

behaviour of the Post model. Among the Eulerian models only the New model is able to

reproduce the initial behaviour of the Post model. The initial phase is controlled by

coalescence phenomenon, so it can be expected that this phase is correctly modeled by the

New model. Then, the collision efficiency is decreasing as far as the Post model is concerned.

This is not captured in the New model although it is able to reproduce the Georjon model.

Comparison of the different assumptions used to derived Georjon and Post models, shows that

the rebound phenomena can be the cause of discrepancy between the New model and its Post

23

counterpart. This could be taken into account through a modification of the probability density

function of the parameter β , see Eq. (28).

(Figure 7: about here)

However, in our opinion, before going further in the integration of collisions features occurring

between spherical droplets, a missing point in the modelling framework must be addressed.

A common drawback of all collision models is the assumption that collisions occur between

spherical droplets whose surface is at rest. But after a collision the droplets are very perturbed

and they are animated by strong oscillations, see Fig.2. It can be expected that collisions lead

to different behaviour depending on the internal agitation of the colliding droplets. The

amount of energy that can be dissipated by the droplet oscillation is shown Fig. 4. Without

any forcing procedure, the kinetic energy of the liquid is strongly reduced. This energy

becomes internal liquid agitation within the droplets, where it is finally dissipated. To recover

a collision between spherical droplets the time needed to dissipate this internal motion must

be shorter that the collision time. To measure the importance of this phenomenon, the Fig. 2

can be observed. After the collision, the two big droplets oscillate nearly all along their ways

out of the measuring zone. Thus the distance covered before the vanishing of the oscillation is

about:

DLdiss 20≈ . (30)

24

To get spherical collision, the characteristic dissipation time dissτ must be smaller than the

collision characteristic time of one droplet (collτ ):

60

1<⇒>= lcolll

dissdiss u

L φττ . (31)

For this typical case, spherical collisions are limited to sprays where the liquid volume

fraction is lower than 2%. Since the collision effects have been expected to be more important

for dense spray, it seems legitimate to focus our study on collisions where agitation within the

droplet is considered. To do so a new numerical test case is put forward in the next section.

5 A numerical test case to study collision in dense spray

An investigation is conducted on the equilibrium state and its characterization through a

Weber number for dense spray. To describe the internal agitation of the droplets, a full DNS

for both the gas and the liquid phase has been used and a first test case is put forward as

reference.

5.1 Direct Numerical Simulation: code description

Thanks to recent developments, Direct Numerical Simulation can be a powerful tool to study

two-phase flows [19, 40]. Indeed, from DNS simulations, statistical information can be

collected in the dense zone of the spray where nearly no experimental data are available.

Furthermore, these simulations are predictive and quantitative. They have been used already

to validate modelling proposal [4]. This is the first objective of the work presented below that

25

focuses on the liquid-gas interface density, but with a fine description of turbulence effects on

the development of the interface.

The numerical method must describe the interface motion precisely, handle jump conditions

at the interface without artificial smoothing, and respect mass conservation. Accordingly a 3D

code was developed by Ménard et al. [19], where interface tracking is performed by a Level

Set method. The Ghost Fluid Method is used to capture accurately sharp discontinuities. The

Level Set and VOF methods are coupled to ensure mass conservation. A projection method is

used to solve the incompressible Navier-Stokes equations that are coupled to a transport

equation for level set and VOF functions.

Level Set methods are based on the transport of a continuous function φ , which describes the

interface between two phases [41, 42]. This function is defined by the algebraic distance

between any point of the domain and the interface. The interface is thus described by the 0

level of the Level Set function. Solving a convection equation allows to determine the

evolution of the interface in a given velocity field V [42]:

0.t

=φ∇+∂φ∂

V (32)

Particular attention must be paid to this transport equation. Problems may arise when the level

set method is developed: a high velocity gradient can produce wide spreading and stretching

of the level sets, such that φ no longer remains a distance function. Thus, a re-distancing

algorithm [41] is applied to keep φ as the algebraic distance to the interface.

To avoid singularities in the distance function field, a 5th order WENO scheme has been used

for convective terms [43]. Temporal derivatives are computed with a third order Runge Kutta

scheme.

26

One advantage of the Level Set method is its ability to represent topological changes both in

2D or 3D geometry quite naturally. Moreover, geometrical information on the interface, such

as normal vector n or curvature κ , are easily obtained through:

φ∇φ∇=n , n⋅∇=φκ )( (33)

It is well known that numerical computation of equation (32) and a redistance algorithm can

generate mass loss in under-resolved regions. This is the main drawback of Level Set

methods. However, to improve mass conservation, two main extensions of the method can be

developed: namely the Particle Level Set [44] and a coupling between VOF and Level Set

[45].

Navier Stokes equations

The Level Set method is coupled with a projection method for the direct numerical simulation

of incompressible Navier-Stokes equations expressed as follows:

2

)(

))(2(

)()(

Tp

t

VVD

DVV

V ∇+∇=∇=∇+∇⋅+∂∂

φρφµ

φρ (34)

0=⋅∇ V (35)

where p is the pressure, ρ and µ are the fluid density and viscosity respectively.

Diffusion is estimated with a 2nd order central scheme. Convective terms are approximated by

5th order WENO scheme to ensure a robust behaviour of the solution. Temporal derivatives

are approximated with an Adams Bashforth algorithm.

27

Poisson equation discretization, with a second order central scheme, leads to a linear system

whose matrix is symmetric and positive definite. Various methods can be derived to solve this

system. According to different authors [46], the MultiGrid method for preconditioning

Conjugate Gradient methods (MGCG) combines Incomplete Choleski Conjugate Gradient

(ICCG) robustness with the multigrid fast convergence rate. Moreover, the MGCG method

greatly decreases computational time compared to the ICCG algorithm.

Discontinuities

The interface is defined by two different phases and discontinuities must be taken into

account for density, viscosity and pressure. Specific treatment is thus needed to describe the

jump conditions numerically.

Two different approaches can be used to represent the above conditions, namely the

Continuum Surface Force (CSF) or the Ghost Fluid Method.

To overcome the smoothing effect of the CSF method, the Ghost Fluid Method (GFM) has

been developed by [47]. The formalism respects jump discontinuities across the interface, and

avoids considering an interface thickness. Discretization of discontinuous variables is more

accurate, and spurious currents in the velocity field are thus much lower than with CSF

methods. This procedure is used to discretize all discontinuous variables, namely density,

viscosity, pressure and viscous tensors [48, 49].

In GFM methods, ghost cells are defined on each side of the interface [49, 50] and appropriate

numerical schemes are applied for jump conditions. As defined above, the interface is

characterized through the distance function, and jump conditions are extrapolated on some

nodes on each side of the interface. Following the jump conditions, the discontinued functions

are extended continuously and then the derivatives are estimated.

28

More details can be found in [50] on implementing the Ghost Fluid Method to solve the

Poisson equation with discontinuous coefficients and obtain solution with jump condition.

Level Set-VOF coupling

A lot of liquid parcels (droplets, ligaments, liquid sheets…) are generated in the primary

break up of a jet and the coupling between the VOF and the Level Set methods is necessary.

The main idea is to take advantage of each strategy: mass conservation from the VOF and fine

description of the interface with the level set and Ghost fluid methods. The numerical method

used here is quite similar to the CLSVOF of Sussman and Puckett [19, 45]. Tin our approach,

the main differences with the CLSVOF lie in the fact that the initial redistancing algorithm is

conserved. In addition, the reconstruction technique is modified to define the interface in a

cell thanks to the Level Set position.

5.2 Test case and preliminary results

To compute the whole interactions between the liquid and the gas phases, the DNS approach

described above is used in the following. Similarly to the previous test cases used for Eulerian

and Lagrangian simulations, a homogeneous spray is considered. The computational domain

is a cubic box with periodic boundary conditions in all directions. Since all scales of the two

phase flow are computed, the numerical effort is more intensive than for the previous test

cases. As a consequence it is difficult to achieve statistical convergence by volume averaging

only. Thus, time averaging is also considered. To do so, a stationary behaviour is required

especially for the turbulent kinetic energy. Consequently, to compensate the continuous

dissipation of the kinetic energy, the turbulence has to be forced. For single phase flows [51]

29

and even dispersed two phase flows [52] spectral methods are generally used to inject kinetic

energy in the largest scales of the flow. However, the spectral behaviour of two phase flows,

which is studied here, is not well known. Accordingly, a simple linear forcing method is

applied here to the liquid-gas mixture to limit artificial hypothesis. Recently, Rosales and

Meneveau [38] have shown that this approach is comparable to spectral methods in the

framework of single phase flows.

Thanks to this method, the mean turbulent kinetic energy reaches a targeted level. The linear

forcing consists in adding one source term to the velocity equations similarly to Eq. 29. This

source term is proportional to the velocity fluctuations through a parameterA . This parameter

is continuously adjusting to sustain the prescribed level of turbulent kinetic energy. Figure 8

represents the temporal evolution of the turbulent kinetic energy considering the complete

mixture: gas and liquid.

Initially, eight droplets are dispatched in the cubic box, one at each corner. The initial droplet

diameter is determined by a given liquid volume fraction. An initial rotational velocity is set

for each droplet; the velocity magnitude is in accordance to the prescribed level of kinetic

energy. The linear forcing of the turbulence is initially deactivated. During the first phase,

starting from instable conditions, the flow becomes turbulent and relaxes gently. This leads to

a decrease off the turbulent kinetic energy at the beginning of the simulation. Then, the

forcing method is activated and the turbulence intensity reaches its prescribed level, see Fig.

8.

(Figure 8: about here)

(Table 2: About here)

30

Properties of the simulation are summarized in table 2. After initialisation, the mean

characteristics of the flow reach a stable state. Pictures of the liquid interface during a

collision are shown in Fig. 9. The collision takes place in the bottom left corner. The figure

represents successive images of the liquid surface from left to right and from top to bottom.

The time interval between two consecutive images is 1.7 ms. At the end of this sequence a

new collision is observed for the same parcel of liquid on the bottom right corner. Clearly

there is not enough time between the two collisions to dissipate the agitation induced by the

first collision. Different liquid structures, more or less tortuous, are observed. The interactions

between the turbulent gas motion and the liquid parcel but also between the liquid parcels

themselves lead to non spherical parcels of liquid. Note that the two phase flow is relatively

dilute since the liquid volume fraction is only 5% (but above the limit of 2% found previously

in Eq. (31) to get a regime of spherical collisions). Only the smallest parcels show a spherical

behaviour, but they represent clearly a very small part of the total amount of liquid. Most

collisions occur between parcels of liquid that cannot be considered as droplets.

(Figure 9: about here)

Accordingly, the droplet diameter that is generally used to build the Weber number of

collision cannot be chosen anymore. Coming back to the formulation proposed in the context

of Eulerian methods, the Weber number of collision writes:

Ω=

Ω= ~

~

3

12~

2

1~

12

2

l

ll

l

ll kYuYWe

σσ (36)

The relative velocity between liquid parcels lu has been expressed in terms of turbulent

kinetic energy in the liquid phase, lk similarly to Eq. (18). The modelling constant appearing

31

in the equation is set to unity. The evolution of the Weber number of collision is represented

in Fig.10. The dashed line represents the instantaneous evolution obtained by considering the

whole computational domain. The solid line represents the collision Weber number averaged

over time.

After the initialisation phase, the Weber number of collision oscillates around its equilibrium

value. The mean value found for this case is about 5.3* =We . This value is not identical to

those previously used for Eulerian or Lagrangian models ( 1512 * <<We ). Before stating

about the accuracy of the equilibrium Weber value obtained by DNS, some issues have do be

addressed. Concerning the influence of the mesh, a test case using a grid of 64x64x64 mesh

cells has been tested and gives similar results. A larger box is expected to get a proper

statistical convergence over the computational model to study also the evolution of the Weber

number. It is also important to note that the periodic boundary condition imposes a scale of

symmetry. To get more universal results the distance between periodic conditions must be

larger than the other scale of the flow, in particular larger than the free mean path of collision.

As a consequence very dilute sprays should be difficult to study using this approach. Despite

these drawbacks the DNS approach brings some light on collision processes for spray not

completely dispersed. It is important to stress that sprays characterized by a liquid volume

fraction of few percents are sprays where collisions are prevalent. The present simulation

demonstrates the importance of collisions between non spherical droplets. More data are

necessary to characterize this phenomenon. The present work shows that comprehensive

results can be obtained thanks to the complete DNS of two phase flows.

(Figure 10: about here)

6 Conclusion

32

Based on recent progresses in the field of atomisation modelling, it appears necessary to study

collisions in moderately dense flows. Special atomisation models able to compute realistically

the dense zone of the spray have been developed since the pioneering work of Vallet et al.

[11, 17]. A difficult point concerns the representation of the collision processes, in particular

on the evolution of the liquid-gas surface density. A description of the ELSA model is

presented to show the part of the model where collisions are expected to play a role. Other

formulations of the Eulerian models dedicated to collisions are compared and the importance

of the equilibrium Weber number has been demonstrated. Classical models of atomisation

have been developed in the context of Lagrangian formulations. These formulations have

been used as reference to test the ability of Eulerian approaches to address the collision

phenomena, in particular coalescence and collision induced breakup. Eulerian models are able

to reproduce correctly the behaviour of simple Lagrangian models of collision [33]. For a

more detailed Lagrangian model [34], the present Eulerian model fails to reproduce the total

complexity of the phenomena. This study shows also that current collision modelling leads to

equilibrium Weber number values ranging between 12 and 15. Before developing more

detailed Eulerian models of collision, it is outlined that current models consider only

collisions between spherical droplets. Experimental observations show that the droplets issued

from a collision are subject to a significant internal agitation, which can play a key role to

determine the outcome of the next collision. When looking at an experimental observation of

a collision, it appears that the droplet oscillations need a long characteristic time to be

dissipated. For this particular test case, a regime of collision between spherical droplets with

no internal agitation can be considered only for spray with liquid volume fraction lower than

2%. To study the influence of non spherical collision in moderately dense spray, a numerical

test case based on complete DNS of the spray has been put forward. Despite some drawbacks

discussed in this paper, the behaviour of the liquid-gas interface for a spray undergoing

33

collisions with stationary turbulent conditions has been studied. As expected, the spray is

mainly composed of non spherical droplets. Collision behaviour looks different to those

encountered when considering collisions of spherical droplets. A first equilibrium Weber

number of collisions has been determined with a value of 3.5. This value differs from those

found previously for Eulerian and Lagrangian models. This may be an effect of non spherical

collisions, but DNS test cases must be firmly established to assess this equilibrium Weber

value definitively. Using a complete DNS lets foresee interesting future prospects to better

understand turbulent two-phase flows. In particular turbulence and collision effects on the

topology of the liquid-gas interface will be studied soon. This in complement to experiments

on non spherical droplet collision is certainly one of the most interesting perspectives of this

work.

7 Acknowledgment

This work was financially supported by PSA Peugeot Citröen. All the authors would like to

thank them for their continuous support and their trust. Particular thanks are given to Pr.

Brenn for his help. The authors thank the CRIHAN and CINES for CPU time and assistance.

34

8 Nomenclature A Turbulence forcing parameter

32D Sauter mean diameter

f Frequency per unit of volume L Characteristic length scale k Turbulent kinetic energy

ELSAvm ,& Vaporisation rate per unit of liquid surface

dN Droplet number per unit of volume

n Vector normal to the liquid surface κ Oh Ohnesorg number P Pressure Sc Schmidt number

effS Cross-section of collision

u Velocity V Velocity field Y Mass fraction We Weber number Greek Symbols

lφ Liquid volume fraction

φ Distance function β Number of droplet issued of a binary collision κ Curvature of the surface Ω Liquid surface density per unit of mass Ψ Repartition function of surface density source terms between dense and dilute zone

collη Coalescence efficiency

ρ Density Σ Liquid surface density per unit of volume σ Surface tension coefficient µ Dynamic viscosity τ Characteristic time scale Subscripts coll Collision diss Dissipation of droplet internal agitation g Gas l Liquid t Turbulent Superscripts * Equilibrium value

35

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Liquid volume fraction 0.1 Mean Sauter diameter (m) 8.9e-05 Liquid kinetic energy (m2.s-2) 1.19 Box Length (m) 0.001 Liquid density (kg.m-3) 991 Liquid tension surface (kg.s-2) 0.07 Table 1: Test case characteristics. Domain sizes (m3) 0.013 Grid 1283 Prescribed turbulent kinetic energy (m2.s-2)

0.08

Liquid volume fraction 0.05 Gas density (kg.m-3) 25 Liquid density (kg.m-3) 753.6 Liquid surface tension (N.m-1) 0.0222 Table 2: Test case characteristics.

39

Figure 1: Temporal evolution of Weber of collision for the three different Eulerian models.

40

Figure 2: Satellite droplet formation by binary drop collisions, [31]

41

Figure 3: Periodic box: Initial field, droplet colorized by the magnitude of their velocity (m.s-

1).

42

Figure 4: Temporal evolution of the liquid kinetic energy with or without forcing.

43

Figure 5: Temporal evolution of the collision Weber number for the three Lagrangian models: O’Rourke, Georjeon and Post.

44

Figure 6: Temporal evolution of the collision Weber number for the three Eulerian models: Iyer, Beau, New compared with the Lagrangian model Georjeon

45

Figure 7: Temporal evolution of the collision Weber number for the three Eulerian models: Iyer, Beau, New compared with the Lagrangian model Post

46

Figure 8: Temporal evolution of the turbulent kinetic energy.

47

Figure 9: Surface evolution during collision process, one frame every 1.7ms, the computational domain volume is V=0.013[m3].

48

.

Figure 10: Temporal evolution of the collision Weber number.